During the repair of turbine blades or vanes, the worn ceramic protective layer has to be removed and, after restoration, reapplied. In this case, the cooling-air holes which are present are partially or completely closed during the coating process. The position and orientation of the borehole axes of the cooling-air holes cannot be determined or can only be partially determined. To date, the holes have been identified in part by finding slight depressions in the ceramic layer and/or relatively small openings, and opened with the aid of a manual process. A reliable, controllable system is not available.